A COUNTEREXAMPLE TO A PROBLEM ON POINTS OF CONTINUITY IN BANACH SPACES

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 99, Number 2, February 1987 A COUNTEREXAMPLE TO A PROBLEM ON POINTS OF CONTINUITY IN BANACH SPACES N. GHOUSSOUB, B. MAUREY AND W. SCHACHERMAYER ABSTRACT. In a previous paper of the first two authors [GM] the space JToo was constructed as a James space over a tree with infinitely many branching points. It was proved that the predual Boo of JToo fails the "point of continuity property." In the present paper we show that Boo has the so-called "convex point of continuity property" thus answering a question of Edgar Wheeler [EW] in the negative. 1. Definitions, notations preliminaries. Recall [BR] that a Banach space X has the point of continuity property (PCP) if for every weakly closed bounded subset C of X there is x 6 C such that the weak norm topology restricted to C coincide at x. An equivalent formulation goes as follows (compare [B, Proposition 1]): For every bounded subset C of X for s > 0 there is a relatively weakly open subset U of C such that the norm-diameter of U is less than Previously, J. Bourgain [B] introduced (under the name of property (*)) the following weaker concept: A Banach space X has the convex point of continuity property (CPCP) if every closed convex bounded subset C of X has a point x where the relative weak norm topologies coincide. Again this may be rephrased as follows: For every convex bounded subset C of X e > 0 there is a relatively weakly open subset U of C of norm-diameter less than e. The question whether PCP CPCP are in fact equivalent remained open was explicitly asked in [EW]. We shall show that the space Poo furnishes an example with CPCP but failing PCP. Recall from [GM] the definition of JT^: We consider the tree with infinitely many branching points = N* fc=0 If t (ni,...,rik) Too, set i = k to be the rank of t. If x (xt)ter00 is a real-valued function of finite support defined on Too, let I JTec = SUp the supremum taken over all families (Si,..., Sn) of disjoint segments in Too- JTa will be the completion of this normed space. Received by the editors December 25, Mathematics Subject Classification (1985 Revision). Primary 46B20, 46B22, 46G American Mathematical Society /87 $ $.25 per page

2 POINTS OF CONTINUITY IN BANACH SPACES 279 As the finite vectors are dense in JT^, an element y G JT is characterized by its values yt = y{et) on the unit vectors et, t G Tx,. The space Poo will be the norm closure of the span of the coefficient functional again denoted {et,t G Too} in IT* We shall say that a subset A Ç Too is full) if for each segment S, the intersection S fi A is again a segment. In this case, the projection is a contraction. Pa- JToo ~* JToo, Yl Xtet ~~* X!Xitt tetx The adjoint of Pa still denoted Pa defines a contraction from Poo to Poq. We shall need the following cases of full sets A Ç Too'- If 7 is a branch in T», we denote by P1 the projection defined by the set 7. Ln will denote the nth line of Too, i-e. Ln: {t: \t\ = n} Pn the projection defined by Ln. lin < m we denote P the projection defined by Ln U Ln+i U U Lm- For each branch 7 = {qb, (ni), (ni,ri2),..., (rii,n2,...,n ),...} in T», define S^.JT^^R, y-^limyt which is well defined as P-^JToo) is isometric to the usual James space J. The collection of branches 7 may be identified in an obvious way with NN. The following lemma is just a slight variant of Theorem 1 of [LS] the proof carries over almost verbatim. 1.1 LEMMA. The operator tea S: JT, - Z2(NN), y^(\imyt) 1 NN is a well-defined quotient map the kernel of S equals Poo- Note the following consequence of Lemma 1.1 which we shall use in the sequel: 1.2 LEMMA. Let y e JT such that (*) liminf Pn(y) oo=0, n >oo where oo denotes the sup-norm for a function defined on T». Then y is an element of Boo- In particular, if (*) holds true, there cannot be an increasing sequence (nfc) L, a > 0 such that M HCíÍ(2/)lljT;>a, fc = l,2,... PROOF. Condition (*) implies that for each branch 7 the limit S^y) equals 0; hence S(y) =0 by Lemma 1.1 we infer that y 6 Pqo- For the second part, note that Poo is spanned by the coefficient functionals et, t G Too, hence (**) is contradictory to (*). G 2. The main result. The following lemma is crucial for the proof of Theorem 2.2. The proof is an interplay of a gliding hump argument with the formation of Cesàro means.

3 280 N. GHOUSSOUB, B. MAUREY AND W. SCHACHERMAYER 2.1 LEMMA. Let C be a convex, bounded subset of Boo, MEN, s > 0. There is a relatively weakly open convex subset U of C N > M such that \\Pn(U)\U = sup{ Pv(y) oo: y G U} < s. PROOF. We may do suppose that C is contained in the unit ball of Poo that 0 < e < 1. Choose natural numbers n m such that n > 2/e m > (8n/e2) + 1 choose n > 0 such that n < s/2mn. Fix an element y in C choose N > M such that Pfv(î/o)ll o < " Suppose the lemma is not true (for the N chosen above). We shall obtain a contradiction by double-induction: For 0 < i < n 1 < j < m we shall find yj G C, for 1 < i < n 1 < j < m, t\ G Ln such that (i) WM)\>, í<j<m,l<i<n, (ii) yí = n-1(y( yi-1), 2<j<m, (iii) \(y - yi)(tqp)\ < n, if (q,p) < (j,i) lexicographically, i.e. q < j or q = j p < i, (iv) lyiitp)] <rl if (<?>P) > Cîi») lexicographically. Let us suppose for the moment that we have done the construction finish the proof. Fix (q,p) with 1 < q < m 1 < p < n. Note that (ii) (iv) imply l?/o(^p)l < V- Now apply (i) (for (j,i) equal to (q,p)), (ii), (iii), (iv) to obtain \yl+\tl)\>e/n-2r,. Repeated application of (ii) (iii) shows Hence \yo(t9p)\ >e/n-(m-q+ l)r} > e/n - mn > e/2n. m 1 n iiciil > n^(y0m)hl > E Eic^)!2 9=1 P=l > (m - 1) n (e/2n)2 > 2 which contradicts the assumption that C is contained in the unit ball of Pc». So, let us do the inductive construction. We have already chosen y ; let Cq C. By assumption there is y\ G C t\ G Ln s.t. yî(<i)l > - ^et A\ = {tgln: \y\{t)\>ri} C\ = {yg Co1: \(y - i/ )(í) < n for t G A1}. Clearly C\ is a relatively weakly open convex subset of C; hence by assumption there is y\ G Cl t\ G Ln s.t. I2/2 (*2)I > e- ^et Al=A U{tGLN: \y%{t)\ > n) C\ = {yg Co1: \(y - y )(*) <»? for t A1}. Continue in an obvious way to find yl,-.,y t\,...,tn satisfying (i), (iii), (iv). Define yl = n-l(y\+-- + yln), Al = Aln,

4 POINTS OF CONTINUITY IN BANACH SPACES 281 Ci = {yg Co1: \(y - y20)(t)\ < r, for t G A20}. Again Cq is a relatively weakly open, nonempty, convex subset of C; hence we may find y2 G C$ t\ G Ln such that 2/2(i2) > s. Note that t2 cannot belong to Aq as the elements of Cq are smaller than 2n + n~1 on the elements of Aq. Hence (iv) is satisfied for (q,p) = (2,1) (i,j) < (q,p)- Let A2 = Alu{tGLN: \yl(t)\>v} C\ = {yg C2: \(y - y2)(t)\ < r, for t G A2}. Again by assumption there is y2 G C\ t2 G Ln s.t. lî/f^i)! > > etc-! we thus find y2,..., y2 t2,..., i2 satisfying (i), (iii), (iv). Now define yl = n-l(yl y2n), A* = A2, Co3 = {V e C02: (i/ - y30)(t)\ <n{ortg Continue in an obvious way for j = 1,..., the proof of the lemma. D 2.2 THEOREM. Poo has CPCP. A3Q}. m to finish the inductive procedure PROOF. If the theorem were false we could find a convex bounded C Ç Poo a > 0 such that each relatively weakly open subset U of C has norm-diameter greater than 3a. Again we shall argue inductively: Let yi G C, \\yi\\ > a find x\ G JToo, \\xi\\ = 1 of finite support, say supp(xi) Ç L\ U U Lm, s.t. (xi,yi) > a. Let Ci = {ygc: (xuy) >a} apply Lemma 2.1 to Ci to find N\ > Mi a relatively weakly open, nonempty, convex D\ Ç Ci s.t. Pjv(-Di) oo < 1- Finally note that Pt1(B00) is isomorphic to I2, which has CPCP, hence we may find a relatively weakly open, nonempty, convex E\ Ç D\ s.t. hence diam(p1nl(pi))<a; diam(p?1+1( ;i)) > 2a. Find t/2 G E\ a finitely valued x<i G JT» with the support contained in LaTí+i U U m, llalli = 1, s.t. ( 2,2/2) > a let C2 = {yge1:(x2,y) > a}. Apply Lemma 2.1 to find N2 > M2 D2 G C2, s.t. Pn2(Ö2) oo < 1/2. Finally find E2 Ç D2 s.t. diam(p1ar2(p2)) < a; hence diam(p^ + 1(P2)) > 2a. Continue in an obvious way to find C D Dn D En D Cn+i D -, Mn < Nn < Mn+1 < Nn+i < -, xn G JT^, \\xn\\ = 1 with supp(xn) Ç LAr _1+i U U LMn s.t. CmQ{yG Boo: (xn,y) > a}, m> n, while Pvn(Cm) oo < n-1, m>n.

5 282 N. GHOUSSOUB, B. MAUREY AND W. SCHACHERMAYER Let Cn be the o-( JT, JTxO-closure of Cn. By weak-star compactness there is an element y G JT, oo ye f C n=l This y has the (impossible) properties described in Lemma 1.2, so we arrive at a contradiction prove the theorem. D References [BR] J. Bourgain H. R Rosenthal, Applications of the theory of semi-embedding s to Banach space theory, J. Funct. Anal. 52 (1983), [B] J. Bourgain, Dentability finite dimensional decompositions, Studia Math. 67 (1980), [E W] G. A. Edgar R. F. Wheeler, Topological properties of Banach spaces, Pacific J. Math. 115 (1984), [GM] N. Ghoussoub B. Maurey, G s-embedding s in Hubert space, J. Funct. Anal. 61 (1985), [LS] J. Lindenstrauss C. Stegall, Examples of separable spaces which do not contain I1 whose duals are not separable, Studia Math. 54 (1975), DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF BRITISH COLUMBIA, VANCOU- VER, B.C., CANADA V6T 1T4 DEPARTEMENT DE MATHÉMATIQUES, UNIVERSITÉ PARIS VII, PARIS, FRANCE INSTITUT FÜR MATHEMATIK, JOHANNES KEPLER UNIVERSITÄT, A-4040 LINZ, AUSTRIA